We estimated the relationship between sea star abundance reported in the shallow nearshore roving-diver surveys and SST by fitting a hierarchical ordinal regression model with a probit link functionEmbedded ImageEmbedded Imagewhere the cumulative probability of the ith survey falling in the jth AC or below is modeled as a function of the following: θj threshold parameters across ACs, which provide a separate intercept for each category j, an SST anomaly metric (SSTmetrici), days since the SST anomaly metric was observed (Days.SSTmetrici), year (Year2007–2017), month (Monthi), and latitude (Latitudei). Year was included as a fixed effect to determine the year when sea star abundances collapsed (years that were statistically significantly different from the 2006 baseline). Month and latitude were included as random effects to account for additional variation over time and space. σ2μ and σ2γ corresponded to the variance of the distribution of month and latitude random effects, respectively. Our data met all model assumptions: (i) the response variable was measured on an ordinal scale; (ii) the predictor variables were continuous or categorical; (iii) there was no multicollinearity among predictor variables, which we assessed with correlation tests for correlations between two predictors and visually for correlations among three predictors; and (iv) there were proportional odds between each AC as indicated by nearly identical effects among generalized logistic regression models comparing each AC split individually (slopes < 2). We fit 10 candidate models that included the year, latitude, and month covariates and one of the following SST metrics: the maximum SST in the 30, 60, 90, 180, or 360 days prior to each roving diver survey; or the maximum anomalous SST in the 30, 60, 90, 180, or 360 days prior to each roving-diver survey. We compared the AIC value of the candidate models with and without the covariate “days since the SST metric was observed,” and then selected the model with the lowest AIC value (tables S1 and S2). We assessed convergence of models by inspecting the maximum absolute gradient of the log-likelihood function and the magnitude of the Hessian. Each model was empirically identifiable by ensuring that the condition number of the Hessian measure was no larger than 104 (47). We evaluated variance explained by the final model using Nagelkerke’s pseudo R2 (31). Nagelkerke’s pseudo R2 is a commonly used statistic to measure goodness of fit that is calculated by comparing likelihood ratios between a full model and an intercept model. We conducted this analysis in R statistical software v3.4.3 (48) using the clmm function of the “ordinal” package (47) for the ordinal regression model and the nagelkerke function in the “rcompanion” package to calculate pseudo R2 values (49).

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